Question 33
A line segment $BC$ is given in the plane. Consider all acute-angled triangles $ABC$ with $|\angle BAC| = 45^\circ$. In each such triangle, denote by $D$ and $E$ those points on the sides $AB$ and $AC$, respectively, such that $BC$ is a common tangent of the circumcircles of triangles $ACD$ and $ABE$. Finally, denote by $P$ and $Q$ the feet of perpendiculars from the points $D$ and $E$ to the line $BC$, respectively. Prove that as points $P$ and $Q$ vary, there exists a fixed point $X$ not on the line $BC$ such that $\angle PXQ$ is fixed.